1461 
8 «= 10—6—8 = — 4, which bears out that the 
formula may not be applied to the circle. In fact £2 consists in this 
case of a twice to be counted equilateral hyperboloid of revolution 
(ef. § 2), and the nodal curve is consequently indetinite. A certain 
control on the general. case we find in the circumstance that the 
order of the rest-nodal curve must be even, as it is, just as the 
surface on which it lies, symmetrical with regard to 8, and must 
therefore be cut by a vertical plane in an even number of points. 
It is true, such a plane contains the point Z, 
dan 
circle 5 u—3 pv 
, which is its own 
image with regard to 8; it will, however, appear that the multi- 
plicity of Z is indicated by an even number, and as the jinite 
intersections on account of their symmetry are also present in an 
even number, the complete ordernumber must be even. This now 
may be proved indeed. 
According to the formulae of Prücker we have: 
“—=td 3 (u 
3u (u—2)—6d—6e (e—1)—Bx '), so 
Su Ou 6d—6e (e—1)—8:— 24u + 24, or 
6d = 3u°—30u—B6e* + Ge Me + 24y, and consequently 
4d = 2u?>—20u—4e* + deb + 16> ; 
if these values are substituted, we find for the order of the rest- 
nodal curve : 
2u? + 4ur—8ue—4uo— Bve + Be? + Beo 13 4 12e + 564 Bu + 
+ vr’ + 6°— Iro + 31; even must therefore be: 
du + de + v?—13r + o? + 56, or 
3 (u + 0) Hv (p—138) + 6 (6 + 5). 
It stands to reason that »(»—13) and o(6+5) are even, and 
further is 
yv) 
t 
t 
i-+ w= 8u? — 5u — 6d — be (e—1) — 8x; 
3u? —5u, or u (du—5) is, however, always even again, so d is after 
all always even. 
The multiplicity of Z, as point of the rest-nodal curve we find 
as follows. According to § 2 there pass through Z, 
1. w--2e— 26 nodal edges (torsal lines) of 2, arising from the 
single intersections of 4” with /,; 
2. 26 nodal edges, lying in pairs infinitely near (also torsal lines) 
arising from the o points of contact of 4e with /, 
Through the edges of the first group pass 2 sheets of £2, touching 
each other along the whole of that edge, while the common 
1) Anw. Cykl. p. 10. 
94* 
